The best place to stay in the city is right downtown, and the best hotel to stay in downtown is the elegant Renaissance Cleveland. Located on the corner of Public Square, this well-established feature of the city offers not only the best location, but the best amenities in the city. It has a large atrium with a great restaurant and an elegant bar for earlier drinks or after-hour cocktails. The rooms are well furnished and exactly what you would expect from a Renaissance. The views from the hotel are also excellent. If you request a room on the north side, you will have a spectacular view of the square and the adjacent BP building (the tallest building in Cleveland). In the distance you will be able to see the lake and, if you lucky, part of the Rock and Roll Hall of Fame.
The hotel is connected to the shopping center of the city (Tower City). In the basement of Tower City, you will also find a Rapid terminal, which provides a very quick and easy way to get to and from the airport and around the city. The hotel is also an easy 5-minute cab ride or a 20-minute walk from almost anything you would want to see in the city, including the Rock and Roll Hall of Fame, the Great Lakes Science Center, The Cleveland Browns Stadium, Jacob's Field, Gund Arena, The Playhouse, The Flats, and the Warehouse District.
The concierge will be more than willing to help you get a cab, as well as give you the schedules and directions to all major, and even some of the minor, events in the city.
The rooms are a little pricey at $190/night and up, but if you have the money and you really want to experience the city, this is a great place to do it.